Golden Germ Window Line Dichotomy

Abstract

Under RH, the continued third-order golden germ has a sharp on-line and off-line zero dichotomy inside the open golden window.

Theorem 1.1 (On-line germ zeros come from the zeta pullback or p = 2, 3).

$$\begin{array}{r}\forall s\in \mathbb{C},p\in \mathrm{Primes}\left(\mathbb{N}\right),\\ x\left(s,p\right):=p^{-s\times {\phi }^2},y\left(s,p\right):=p^{-s\times {\phi }^3},\mathrm{Kp}\left(s,p\right):=(1-y{\left(s,p\right)}^2{)}^{-1}\times (1-x{\left(s,p\right)}^2\times y\left(s,p\right))\times (1-y\left(s,p\right))\times (1+x\left(s,p\right){)}^{-1}\times \mathrm{germLocalFactor}\left(s,p\right),\\ \forall s\in \mathbb{C},G3\left(s\right):=\prod _{p\in \mathrm{Primes}\left(\mathbb{N}\right)}\mathrm{Kp}\left(s,p\right),\\ RiemannHypothesis\Rightarrow \left(\forall continuedGerm\in \left\{\frac{1}{{\phi }^5}<\Re s\mid s\in \mathbb{C}\right\}\to \mathbb{C},\left(\forall s\in \left\{\frac{1}{{\phi }^5}<\Re s\mid s\in \mathbb{C}\right\},continuedGerm\left(s\right)=\mathrm{riemannZeta}\left({\phi }^2\cdot s\right)\cdot \mathrm{riemannZeta}\left({\phi }^3\cdot s\right)\cdot \mathrm{riemannZeta}{\left(2\cdot {\phi }^2\cdot s\right)}^{-1}\cdot \mathrm{riemannZeta}{\left(2\cdot {\phi }^3\cdot s\right)}^{-1}\cdot \mathrm{riemannZeta}\left(\left(2\cdot {\phi }^2+{\phi }^3\right)\cdot s\right)\cdot G3\left(s\right)\right)\Rightarrow \left(\forall s\in \left\{\frac{1}{{\phi }^5}<\Re s\mid s\in \mathbb{C}\right\},\Re s=\frac{1}{2\cdot {\phi }^2}\Rightarrow \left(continuedGerm\left(s\right)=0\iff \left(\mathrm{riemannZeta}\left({\phi }^2\cdot s\right)=0\vee \left(\mathrm{germLocalFactor}\left(s,2\right)=0\vee \mathrm{germLocalFactor}\left(s,3\right)=0\right)\right)\right)\right)\right).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Analytic/GermWindow/GoldenGermWindowLineDichotomy.golden_continued_germ_line_zero_iff_of_rh (✓ std3). ∎

Source. Repository-derived.

Commentary.

On the pulled-back critical line, the frozen open-window classification and the critical-line nonvanishing theorem exclude every local factor at primes at least five. The remaining alternatives are the phi-squared zeta pullback and the local factors at p = 2 or p = 3.

Whether either small-prime local factor actually vanishes is a numerical question: this theorem neither asserts nor excludes such zeros. The result assumes RH and is not an RH proof path.

Theorem 1.2 (Off-line window zeros are exactly local-factor zeros).

$$\begin{array}{r}\forall s\in \mathbb{C},p\in \mathrm{Primes}\left(\mathbb{N}\right),\\ x\left(s,p\right):=p^{-s\times {\phi }^2},y\left(s,p\right):=p^{-s\times {\phi }^3},\mathrm{Kp}\left(s,p\right):=(1-y{\left(s,p\right)}^2{)}^{-1}\times (1-x{\left(s,p\right)}^2\times y\left(s,p\right))\times (1-y\left(s,p\right))\times (1+x\left(s,p\right){)}^{-1}\times \mathrm{germLocalFactor}\left(s,p\right),\\ \forall s\in \mathbb{C},G3\left(s\right):=\prod _{p\in \mathrm{Primes}\left(\mathbb{N}\right)}\mathrm{Kp}\left(s,p\right),\\ RiemannHypothesis\Rightarrow \left(\forall continuedGerm\in \left\{\frac{1}{{\phi }^5}<\Re s\mid s\in \mathbb{C}\right\}\to \mathbb{C},\left(\forall s\in \left\{\frac{1}{{\phi }^5}<\Re s\mid s\in \mathbb{C}\right\},continuedGerm\left(s\right)=\mathrm{riemannZeta}\left({\phi }^2\cdot s\right)\cdot \mathrm{riemannZeta}\left({\phi }^3\cdot s\right)\cdot \mathrm{riemannZeta}{\left(2\cdot {\phi }^2\cdot s\right)}^{-1}\cdot \mathrm{riemannZeta}{\left(2\cdot {\phi }^3\cdot s\right)}^{-1}\cdot \mathrm{riemannZeta}\left(\left(2\cdot {\phi }^2+{\phi }^3\right)\cdot s\right)\cdot G3\left(s\right)\right)\Rightarrow \left(\forall s\in \left\{\frac{1}{{\phi }^5}<\Re s\mid s\in \mathbb{C}\right\},\frac{1}{2\cdot {\phi }^3}<\Re s\Rightarrow \left(\Re s<\frac{1}{{\phi }^2}\Rightarrow \left(\Re s\ne \frac{1}{2\cdot {\phi }^2}\Rightarrow \left(continuedGerm\left(s\right)=0\iff \left(\exists p\in \mathrm{Primes}\left(\mathbb{N}\right),\mathrm{germLocalFactor}\left(s,p\right)=0\right)\right)\right)\right)\right)\right).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Analytic/GermWindow/GoldenGermWindowLineDichotomy.golden_continued_germ_off_line_zero_iff_of_rh (✓ std3). ∎

Source. Repository-derived.

Commentary.

Away from the pulled-back critical line, the zeta-zero branch of the frozen window criterion is impossible. Thus a continued germ zero in the open window is exactly a zero of some canonical golden local factor.

This is a conditional consequence of RH and the frozen third-order factorization. It supplies no converse and is not a route to proving RH.

References

• Truth anchor: D5/S3/Analytic/GermWindow/GoldenGermWindowLineDichotomy.golden_continued_germ_line_zero_iff_of_rh
• Truth anchor: D5/S3/Analytic/GermWindow/GoldenGermWindowLineDichotomy.golden_continued_germ_off_line_zero_iff_of_rh
• Dependency: D5/S3/Analytic/EulerGerm/LocalFactorCriticalLineNonvanishing
• Dependency: D5/S3/Analytic/GermWindow/GoldenGermThirdOrderWindowZeroCriterion